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Foundations · Numbers

Decimals

Decimal place value, the four operations with decimals, and switching between fractions, decimals and percentages.

First, the rules

Fractions with a denominator of 10, 100, 1000 …

0.7 is seven tenths, 0.07 is seven hundredths. The decimal point just marks where the whole units end — so adding lines the points up, and multiplying counts the places.

★ Places after the point
Example 1
0.472 = 410 + 7100 + 21000 = 4721000
★ Add / subtract
Example 2
Line up the decimal points, fill empty places with 0, then work as with whole numbers: 3.7 + 0.85 = 3.70 + 0.85 = 4.55.
× and ÷ by 10, 100, 1000
Example 2
Move the point: 2.45 × 100 = 245; 2.45 ÷ 10 = 0.245.
★ Multiply
Example 2
Ignore the points, multiply, then give the answer as many decimal places as both factors had together: 1.2 × 0.35 → 12 × 35 = 420 → 3 places → 0.420 = 0.42.
Divide
Example 2
Make the divisor whole by moving both points the same way: 4.5 ÷ 0.15 = 450 ÷ 15 = 30.
★ Fraction ↔ decimal ↔ percent
Example 3
38 = 3 ÷ 8 = 0.375 = 37.5%
0.45 = 45100 = 920 · 13 = 0.333… (recurring)

See it

Zoom into the number line

Each line below cuts one piece of the line above into ten. Slide the value and watch which cut each digit after the point names.

Step by step

Worked examples

1

Ordering decimals

place value

Put in order from smallest: 0.6, 0.56, 0.605, 0.65.

Ask yourself: Give every number the same number of decimal places first.

Show the solution
  1. Three places: 0.600, 0.560, 0.605, 0.650.
  2. Now compare like whole numbers: 560 < 600 < 605 < 650.
  3. So 0.56 < 0.6 < 0.605 < 0.65 — longer does not mean bigger.

Answer 0.56 < 0.6 < 0.605 < 0.65

2

The four operations

operations

Work out (a) 12.6 − 3.85, (b) 0.4 × 2.35, (c) 7.2 ÷ 0.08.

Ask yourself: (a) line up the points, (b) count places, (c) make the divisor whole.

Show the solution
  1. (a) 12.60 − 3.85 = 8.75.
  2. (b) 4 × 235 = 940; one place + two places = three places: 0.940 = 0.94.
  3. (c) Move both points two places: 720 ÷ 8 = 90.

Answer 8.75 · 0.94 · 90

3

Three forms of the same number

conversions

Write 78 as a decimal and a percentage, and 0.325 as a fraction in lowest terms.

Ask yourself: Division for the first; thousandths for the second.

Show the solution
  1. 7 ÷ 8: 8 into 70 goes 8 (64) r 6, into 60 goes 7 (56) r 4, into 40 goes 5: 0.875 = 87.5%.
  2. 0.325 = 3251000; divide by 25: 1340.

Answer 0.875 · 87.5% · 1340

Practice: decimals

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Adding, subtracting, multiplying and dividing decimals, moving the point, and converting to fractions and percentages. Your best score is kept in this browser only.

About this lesson

Decimals for school mathematics: tenths, hundredths and thousandths, lining up the point to add and subtract, counting decimal places when multiplying, moving the point to divide, and converting between fractions, decimals and percentages. A number-line zoom you can move, worked examples and practice questions. Everything on this page runs in your browser; nothing you type is sent or stored. Spotted an error? Tell us.